Cassini-Catalan Determinants via Ramanujan's Theta Identity

In this paper, we show that the classical Cassini and Catalan identities for Fibonacci numbers arise naturally from a single quadratic theta-function identity of Ramanujan. Expanding the identity $\psi (q)\psi (q^3)=\psi (q^4)\phi (q^6)+q\phi (q^2)\psi (q^{12})$ via the Jacobi triple product and equating coefficients yields the unified $q$-determinant $F_{n+r}(q)F_{n-r}(q)-F_n(q{)}^2=(-q{)}^{n-r}{F}_r(q{)}^2$, $n\ge r\ge 1$, where $\psi (q)$ and $\phi (q)$ are Ramanujan’s theta functions with $q$ a complex parameter in the unit disc $(\mid q\mid <1)$ and $F_n(q)$ denotes the Carlitz $q$-Fibonacci polynomials. The radial limit $q\to 1^{-}$ recovers Cassini’s formula ($r=1$) and Catalan’s one-parameter extension, while the same derivation with an auxiliary weight produces new partition-refined versions. The argument uses only standard $q$-series algebra (triple-product expansions, $q$-Pochhammer cancellations, and coefficient extraction), providing a transparent modular explanation of the alternating sign $(-1{)}^{n-r}$ in Catalan’s identity through the level-6 provenance of $\phi$ and $\psi$. Beyond unifying Cassini\textendash Catalan in a single framework, the method lifts seamlessly to higher-order recurrences, giving a template for Tribonacci-type determinants and suggesting congruence phenomena obtained from modular dissections and root-of-unity limits. The results place familiar Fibonacci determinants within Ramanujan’s analytic landscape, indicate routes to combinatorial bijections that mirror the analytic cancellations, and connect with themes in modern $q$-series\textemdash ranging from colored partition identities to quantum-modular and exactly solvable models\textemdash thereby highlighting both the explanatory power and the ongoing relevance of Ramanujan’s theta identities.